Continuity

I must say if the function is continuous in the point (0,0). Which is

The function:

I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.

What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.

I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.

What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.

is bounded, so the limit does exists and equals..., and thus the function isn't continuous at the origen.